e_213131
plain-language theorem explainer
Pointwise identity: the folded Regge numerator m2Num at multi-index (2,1,3,1,3,1) equals eight times the closed-form kernel table explicitZ at the same indices. Gravity analysts cite it as one of the 256 kernel decides that assemble the global m2Num = 8·explicitZ statement. The proof is a single decidability check on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,1,3,1,3,1)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(2,1,3,1,3,1)$ equals $8\,Z(2,1,3,1,3,1)$, where $Z$ is the explicit integer kernel table.
background
This module is chunk 9 of a 256-case kernel certification for the 4D Regge exact-midpoint $M_2$ TT identity. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution contrib t a b c d i j. The comparison target explicitZ is a pattern-matched integer table on six Fin 4 indices (sample entries include $4$ on diagonal-type patterns and $-2$ on crossed patterns).
The local claim is purely combinatorial: evaluate both sides at one concrete six-tuple and check equality in $\mathbb{Z}$. No continuum limit or curvature hypothesis enters; the setting is finite-index arithmetic supporting a later global identity over all of $(\mathbb{F}_4)^6$.
proof idea
One-line proof by decide. Lean reduces both m2Num 2 1 3 1 3 1 (the fold over couplingZList) and 8 * explicitZ 2 1 3 1 3 1 (table lookup times eight) to concrete integers and discharges equality by the decidable instance on Int. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases on all six indices. Each chunk theorem such as this one is a leaf of that case split: without the pointwise decides, the global numerator-to-kernel identity does not close.
In the Gravity analysis stack this identity is infrastructure for the Regge exact-midpoint $M_2$ TT certification in 4D, not a direct forcing-chain (T0–T8) step. It locks the discrete kernel arithmetic so later curvature or continuum arguments can quote a single closed form instead of a fold.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.